What is the word that classifies softness/hardness?

First of all, I was wondering, what is the word that classifies softness/hardness? Such as Color is for red/green. Now off to the main point... What makes something soft or hard? Lets take rubber for example. I thought the closer the atoms, the harder the rubber. But then, the more we stretch it, the harder it becomes. But in that case, the atoms are more spread apart. On the other hand, if we bring atoms closer together than the bonds are, it will become softer. I am seriously missing some points as in this case, two principles clash together.

You are almost there - it depends on how strongly the atoms are held together and the arrangement of atoms which controls what bonds they can make to each other.
So both diamond and graphite (coal dust / or pencil ) are both carbon.
But in diamond each carbon atom has a strong bond to 4 other carbon atoms in a pyramid, because of the triangle shapes it is very difficult to move the carbon atoms around and so diamond is very hard.
In graphite the carbon atoms are bonded to each weakly other in flat sheets and there are only very weak bonds between the sheets, so they can slide over each other easily and graphite is very soft and slippery.